Animated Solution for Physics - Thermodynamics: The amount of heat needed to raise the temperature of 4 moles of a rigid diatomic gas from $0^{\circ }\text{C}$ to $50^{\circ }\text{C}$ when no work is done is ......... . ($R$ is the universal gas constant)

A Locked Piston

Imagine you are observing a sealed container filled with $4$ moles of a rigid diatomic gas. The container is fitted with a piston, but there is a catch—the piston is completely fixed. This means no matter how much we heat the gas, it cannot expand. The volume remains strictly constant. Our goal is to find out exactly how much heat is required to raise the temperature of this gas from $0^{\circ }\text{C}$ to $50^{\circ }\text{C}$.

To solve this, we turn to the fundamental principle of energy conservation: the First Law of Thermodynamics. The law states that the heat supplied to a system ($\Delta Q$) is used to do two things: change its internal energy ($\Delta U$) and perform work on the surroundings ($\Delta W$). Mathematically, this is written as:

However, because our piston is locked in place, the gas cannot expand. In physics, work done by a gas is given by $P\Delta V$. Since the change in volume ($\Delta V$) is zero, the work done ($\Delta W$) is also zero. Therefore, the equation simplifies beautifully:

This tells us that every single joule of heat we pump into the container goes directly into increasing the internal kinetic energy of the gas molecules.

Now, we need to calculate this change in internal energy. The formula for the change in internal energy of an ideal gas is:

Here, $n$ is the number of moles, $C_V$ is the molar heat capacity at constant volume, and $\Delta T$ is the change in temperature. The problem specifies that we are dealing with a rigid diatomic gas. The word "rigid" is crucial here. It means the bond between the two atoms is stiff, so the molecule cannot vibrate. It can only translate in $3$ directions and rotate in $2$ independent axes. This gives it $5$ degrees of freedom ($f=5$).

The molar heat capacity at constant volume is directly related to the degrees of freedom by the relation $C_V=\frac{f}{2}R$. Substituting $f=5$, we get:

We now have all the pieces of the puzzle. Let's list our known values: - Number of moles, $n=4$ - Molar heat capacity, $C_V=\frac{5}{2}R$ - Change in temperature, $\Delta T=50^{\circ }\text{C}-0^{\circ }\text{C}=50\text{ K}$

Substitute these into our master equation:

Let's do the math. The $4$ in the numerator and the $2$ in the denominator simplify to $2$. Multiplying $2$ by $5$ gives $10$. Finally, $10$ multiplied by $50$ yields $500$.

And there we have it! The total amount of heat needed is $500R$. By carefully analyzing the physical constraints (no work done) and the nature of the gas (rigid diatomic), a seemingly complex thermodynamics problem unravels into a straightforward calculation.